The essential element for the territories is whether the federal US has jurisdiction there in addition to sovereignty. The US definitely has sovereignty over American Samoa, as it is a territory not a protectorate, but the jurisdiction remains in the local entity of American Samoa, under its constitution and statutes, except where it has been explicitly ceded or shared, or seized.
As such, the court would have to examine whether he was subject to the jurisdiction of the United States at the time he was born. As the United States makes a distinction between "incorporated" and "unincorporated" territories, one could argue that the "unincorporated" status is an indicator that the US refuses to extend its jurisdiction there.
So he does not have a valid claim under the 14th Amendment. It would be up to American Samoa to petition Congress to be "incorporated", and thereby subject itself to the federal jurisdiction of the US, and also to make that status retroactive beyond the birthdates of the class plaintiffs, in order for the court decide the case has merit. If it were to do so, the exclusionary local laws that favor island natives would then be subject to examination under equal protection and federal nondiscrimination clauses. If they want equal rights as the mainlanders, they have to erase the distinctions on the island, too.
If Samoa becomes incorporated, all the normal constitutional protections apply.